NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes

Abstract

We address fundamental challenges in representing and computing ${\mathbb{R}}^d$-valued predictable square-integrable processes over $[0,T]$, collected in the space ${\mathcal{H}}_T^2({\mathbb{R}}^d)$. These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although Wiener-chaos expansions offer strong theoretical tools, traditional computational methods are hindered by the need for large chaos dictionaries and high-order iterated integrals. To overcome these obstacles, we introduce NeuralChaos -- a neural operator architecture that produces elements of ${\mathcal{H}}_T^2({\mathbb{R}}^d)$ using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in ${\mathcal{H}}_T^2({\mathbb{R}}^d)$ and achieves the best $N$-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from ${\mathcal{H}}_T^2({\mathbb{R}}^d)$ under non-degenerate sub-Gaussian sampling. In contrast, we show that finite-dimensional Markovian neural SDE models constitute a meagre and Gaussian-null subset in ${\mathcal{H}}_T^2({\mathbb{R}}^d)$, regardless of discretization, whereas compressible processes are generic. Numerical experiments on a stochastic optimal control problem and dynamic hedging highlight the practical effectiveness of our approach. Our results enable more efficient and expressive modelling in stochastic analysis and mathematical finance.

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